Equations from The Relativistic Transverse Doppler Effect at Distances from One to Zero Wavelengths. Copyright 2006 Joseph A.

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1 Equtins m Th Rltiisti Tnss ppl Et t istns m On t Z Wlngths Cpyight 006 Jsph A. Rybzyk Psntd is mplt list ll th qutin usd in did in Th Rltiisti Tnss ppl Et t istns m On t Z Wlngths pp. Als inludd ll th illusttins usd in th iginl wk.. Fundmntl Rltinships btwn th Emittd ttd W Th gmtil ltinships btwn tw sussi ws t th tim missin by su in unim mtin illusttd in Figu. dtt s θ s θ θ su FIGURE Emittd ttd Wlngth Rltinships. Th Clssil Tnss ppl Et Initil Wlngths Ring t Figu th gmtil ltinships th initil wlngths mittd by ming su uth liid. As n b sn in th illusttin, th ppl td distn dpnds n th ngl btwn th pjtd (dttd lin) ppgtin pth th nxt t b mittd w th pth mtin th su. dtt dtt dtt s θ θ s θ θ θ θ θ θ s s su su su su dtt Rssin Rssin 90 gs Apph A B C FIGURE Clssil Tnss ppl Et ind

2 Sin θ θ supplmnty ngls, thi ltinship t h th is gin by θ = 80 () Rltinship Angl Apph θ t Angl Rssin θ θ θ = 80 () Rltinship Angl Rssin θ t Angl Apph θ θ wh, lthugh ith n b usd t din th ngl th sm pth btwn th su dtt, it is ustmy t us th n tht lls in th ng 0 θ 90. Fm th Lw Csins w thn h = + s θ (3) Initil Wlngth Rltinships w is th spd light, is th spd th su, is th ppl td distn, θ is th ngl ssin. Sling distn thn gis = s θ θ s + + (4) ppl Etd Initil Wlngth th ight sid whih n b pld in d t gi in tms t di ( i) s θ + = s θ + (5) Clssil Tnss ppl Et Ft Initil Wlngth wh (i) is dind s th lssil tnss ppl t t initil wlngths m ding su. Th mplt lssil tnss ppl t initil wlngths mittd by ding su in unim mtin thn is s θ + s θ + = (6) Clssil Tnss ppl Et Initil Wlngth wh is th bsd initil wlngth m ding su, is th mittd pimy wlngth, is th spd th su, θ is th ngl ssin, is th spd light in mpty sp. This mul gis th t initil wlngth th nti ng ngls m θ = O us, t 90 th sult is nith tht ssin n pph, t gt thn 90 th sult is tht pph. Als, by hnging th sign wh shwn, whil hnging th ngl t θ w gt s θ + s θ + = (7) Clssil Tnss ppl Et Initil Wlngth

3 wh th dttd initil wlngth is bsd n th ngl pph θ instd th ngl ssin θ. Agin th sults lid th nti ng th piusly dsibd nditins with th xptin tht sults ngls gt thn 90 nw pply t th nditin ssin nt pph. F nnin, Equtins (6) (7) n b mbind int singl mul initil wlngths. This gis ± s θ + s θ + = (8) Clssil Tnss ppl Et Initil Wlngth wh is th bsd initil wlngth, is th mittd initil wlngth, θ is th ngl bstin, is th spd light in uum,, th spd th su in unim mtin is + ssin pph wh inditd. 3. Th Rltiisti Tnss ppl Et Initil Wlngths F th ltiisti sins th qutins, w simply nd t di x m thn substitut th sult int th piusly did qutins thn simpliy thm s nssy. T di x m w simply t by th millnnium thy quilnt th Lntz gmm t t btin x = (9) Wlngth Tnsmtin - Unim Mtin Fm t Sttiny Fm wh x is th tnsmd initil wlngth tully td n by th piusly did lssil tnss ppl t t (i) m Equtin (5). Substituting x in pl in th just di qutins (6), (7), (8) thn substituting th ight sid Equtin (9) x llwd by simpliitin thn gis s θ s + θ + = (0) Rltiisti Tnss ppl Et, Initil Wlngth s θ s + θ + = () Rltiisti Tnss ppl Et, Initil Wlngth ± s θ s + θ + = () Rltiisti Tnss ppl Et, Initil Wlngth wh θ in Equtin (0) th spti initil pimy wlngth ngl dttin ssin, θ in Equtin () th spti initil pimy wlngth 3

4 ngl dttin pph, θ in Equtin () th spti initil pimy wlngth ngl dttin ssin pph wh in th ist tm n th tp th tin is + ssin pph. Sin th ltinships quny t wlngth gin by = (3) Obsd Fquny t Obsd Wlngth Rltinship = (4) Emittd Fquny t Emittd Wlngth Rltinship w n substitut th ight sids ths qutins in pl sptily in Equtin () t btin = ± s θ + s θ + (5) tht upn simpliitin gis = ± s θ + s θ + (6) Rltiisti Tnss ppl Et, Initil Fquny wh is th bsd initil quny in th sttiny m, is th mittd initil pimy quny in th ming m, is + ssin pph wh inditd. 4. Th Clssil Tnss ppl Et t Ftinl Wlngth istns Ring t Figu 3, th initil wlngth illusttd in iw Figu is shwn gin, but with th dditinl tus ssitd -lbling ndd tinl wlngth dttin distn nlysis. In gd t tinl wlngth dttin distns, th wlngth dttd t pint, whn th w m th psnt ltin th su tls distn d, is gin by th mul d (7) ttd Wlngth t Pint = + wh is th wlngth dttd t pint tinl wlngth distn d. istn is ltd t distn by th mul = + (8) Rltinship Ftinl Wlngth istns 4

5 = (9) Ftinl Wlngth istn s sn in Figu 3. dtt = + d θ s θ θ s su FIGURE 3 Ftinl Wlngth istns Th mliz ltinship btwn ngl ssin θ tinl wlngth distns, its supplmnty ngl pph θ is gin by θ = 80 (0) Rltinship Angl Apph θ t th Angl Rssin θ θ θ = 80 () Rltinship Angl Rssin θ t th Angl Apph θ θ wh ith n b usd t din th ngl th sm pth btwn th su dtt, lthugh it is ustmy t us th n tht lls in th ng wh 0 θ 90. F distn whn su spd, distn d, ngl dttin θ knwn w gin st t th Lw Csins btining = + d d s θ () Fm Lw Csins Thm thus giing = + d d s θ (3) Ftinl Emittd Wlngth istn wh is th tinl wlngth distn btwn pint s pint, is th spd th su, d is th tinl wlngth distn btwn th su th pint dttin, θ is th ngl dttin. Ring bk t Equtin (9) w n nw stt by wy substitutin = + d d s θ (4) Ftinl Emittd Wlngth istn giing 5

6 x d + = (5) Ttl Wlngth istn ttd t Pint wh x is th ttl wlngth distn dttd t pint. Thugh substitutin with Equtin (4) this gis x = d + + d d s θ (6) Ttl Wlngth istn ttd t Pint wh distn x n b ntd t t in tms thugh diisin by, giing x( ) d + + d d s θ = (7) Clssil Tnss ppl Et Ft Ftinl Wlngth istns wh x() is th quid t th lssil tnss ppl t tinl wlngth distns. F th lssil tnss ppl t tinl wlngth distns thn, this gis d + + d d s θ = (8) Clssil Tnss ppl Et Ftinl Wlngth istns wh is th dttd initil wlngth t pint m ding su, is th mittd pimy initil wlngth, d is th tinl wlngth distn dttin, is th spd th su, θ is th ngl ssin, is th spd light in mpty sp. Exping n this s dn th initil wlngth muls did li gis d + + d + d s θ = (9) Clssil Tnss ppl Et Ftinl d + + d ± d s θ Wlngth istns = (30) Clssil Tnss ppl Et Ftinl Wlngth istns wh is th initil wlngth dttd t pint t n ngl pph θ, is th initil wlngth dttd t pint t n ngl dttin, θ. As with th initil wlngth muls, hng sign is quid in ths lst th qutins inling tinl wlngth distns. In this s, hw, th signs sd; i.. ssin + pph tk pl insid th dil s shwn. 6

7 5. Th Rltiisti Tnss ppl Et t Ftinl Wlngth istns Sin it is nt tully th mittd pimy wlngth tht is ptd n by th lssil tnss ppl t t w gin h t substitut th tnsmd wlngth x th mittd pimy wlngth in th just id t muls. And sin, s shwn piusly x = (9) Wlngth Tnsmtin - Unim Mtin Fm t Sttiny Fm w nd t simply t Equtins (8) (9) (30) with th millnnium sin th Lntz tnsmtin t t i t d + = + d d s θ (3) Rltiisti Tnss ppl Et Ftinl Wlngth istns d + + d + d s θ = (3) Rltiisti Tnss ppl Et Ftinl Wlngth istns d + + d ± d s θ = (33) Rltiisti Tnss ppl Et Ftinl Wlngth istns wh,, th spti wlngths dttd t pint m ding, pphing, ith typ su, is th mittd pimy wlngth in th su s m n, d is th distn btwn th su dtt t th instnt missin, θ, θ, θ th spti ngls dttin, is th spd th su, is th spd light in uum. (Nt sign hng in qutin (33); ssin + pph wh inditd) Agin, lthugh ny ngl m 0 t 80 n b usd in ny ths muls, it is ustmy t us th mul wh th ngl lls in th ng 0 θ 90. F th quny sin th mul w n, s dn piusly th initil wlngth mul, pply th ltinships gin by qutin (3) t qutin (33) t i t = (34) Rltiisti Tnss ppl Et Ftinl d + + d ± d sθ Wlngth istns 7

8 wh is th quny dttd t pint m ding pphing su, is th mittd quny, th tm insid th dil ntining θ is ssin + pph s disussd piusly. 6. Th 90 Rltiisti Tnss ppl Et t Ftinl Wlngth istns F th 90 tnss ppl t usd in mny xpimnts ndutd t tinl wlngth distns m th su, muh simpl mul is ilbl using th Pythgn Thm. Figu 4 shws simpliid sin th initil wlngth piusly illusttd in Figu 3. Nw, hw, th ngl dttin btwn th su dttin pint is limitd t 90 s shwn. Thus, th tinl wlngth distn piusly lbld is nw ditly dind by th Pythgn Thm in tms su spd dttin distn d. = + + d dtt + d d θ s s su FIGURE 4 90 Ftinl Wlngth istns Pding s b, th lssil tnss ppl t dttin pint is gin by x = d + (5) Ttl Wlngth istn ttd t Pint wh x is th ttl wlngth distn dttd t pint. Sin, in dn with Figu 4 th = + (35) Initil Wlngth btwn Pint s Oiginl tt + d = + (36) Ftinl Emittd Wlngth istn d by wy substitutin th ight sid this qutin int qutin (5) w btin p + = d + d (37) Ttl Wlngth istn ttd t Pint wh x, nw lbld p t id nusin with its th us, is th wlngth dttd t pint. iisin th ight sid th sulting qutin (37) by thn gis 8

9 p ( ) d + + d = (38) 90 Clssil Tnss ppl Et Ft Ftinl Wlngth istns wh p() is th quid t th 90 lssil tnss ppl t tinl wlngth distns. Applying this t t th mittd pimy wlngth whn th su is t 90 ngl t dttin pint gis d + + d = (39) 90 Clssil Tnss ppl Et Ftinl Wlngth istns wh is th wlngth dttd t pint. Fting this qutin with th millnnium sin th Lntz gmm t s dmnsttd piusly thn gis d + + d = (40) 90 Rltiisti Tnss ppl Et Ftinl Wlngth istns th ltiisti sin th 90 mul. Equtins m Th Rltiisti Tnss ppl Et t istns m On t Z Wlngths Cpyight 006 Jsph A. Rybzyk All ights sd inluding th ight pdutin in whl in pt in ny m withut pmissin. Nt: I yu ntd this pg ditly duing sh, yu n isit th Millnnium Rltiity sit by liking n th Hm link blw: Hm 9

Equations from Relativistic Transverse Doppler Effect. The Complete Correlation of the Lorentz Effect to the Doppler Effect in Relativistic Physics

Equations from Relativistic Transverse Doppler Effect. The Complete Correlation of the Lorentz Effect to the Doppler Effect in Relativistic Physics Equtins m Rltiisti Tnss ppl Et Th Cmplt Cltin th Lntz Et t th ppl Et in Rltiisti Physis Cpyight 005 Jsph A. Rybzyk Cpyight Risd 006 Jsph A. Rybzyk Fllwing is mplt list ll th qutins usd in did in th Rltiisti